Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

Previous Article
A two-species weak competition system of reaction-diffusion-advection with double free boundaries
DCDS-B Home
This Issue
Next Article
Mean field model for collective motion bistability

February 2019, 24(2): 831-849. doi: 10.3934/dcdsb.2018209

Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

Dan Li ^1,, , Chunlai Mu ^1, , Pan Zheng ^2, and Ke Lin ^3,

1.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.	Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3.	College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author: liwadanj@126.com

Received October 2017 Revised March 2018 Published June 2018

Fund Project: The first author is partially supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB 17040). The second author is partially supported by NSFC (Grant Nos. 11771062, 11571062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Science Foundation of China (Grant Nos. 11601053, 11526042). The fourth author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059)

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source

$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$

in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills

$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$

with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.

Keywords: Chemotaxis- Stokes, nonlinear diffusion, tensor-valued sensitivity, boundedness, logistic source.

Mathematics Subject Classification: Primary: 35K45; Secondary: 35B65, 35Q92, 35Q35, 92C17.

Citation: Dan Li, Chunlai Mu, Pan Zheng, Ke Lin. Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 831-849. doi: 10.3934/dcdsb.2018209

References:

[1]	K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027.
[2]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[3]	X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913. doi: 10.1088/0951-7715/27/8/1899.
[4]	T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.
[5]	T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045.
[6]	T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004.
[7]	T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.
[8]	K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[9]	K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045.
[10]	M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
[11]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[12]	T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 281-301. doi: 10.1006/aama.2001.0721.
[13]	D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[14]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.- Ver., 105 (2003), 103-165.
[16]	S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2d bounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463.
[17]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
[18]	S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569.
[19]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[20]	A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858.
[21]	A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879.
[22]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[23]	H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106.
[24]	T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746. doi: 10.1142/S0218202515500177.
[25]	X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910. doi: 10.4310/CMS.2016.v14.n7.a5.
[26]	X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30. doi: 10.1016/j.nonrwa.2017.02.005.
[27]	J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028.
[28]	J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999. doi: 10.1016/j.jde.2016.03.030.
[29]	T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[30]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[31]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[32]	Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Zeitschrift f/"ur angewandte Mathematik und Physik, 68 (2017), p68. doi: 10.1007/s00033-017-0816-6.
[33]	Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
[34]	Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift f/"ur angewandte Mathematik und Physik, 67 (2016), p138. doi: 10.1007/s00033-016-0732-1.
[35]	Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y.
[36]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[37]	Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.
[38]	Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.
[39]	M. Winkler, Global weak solution in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.
[40]	I. Tuval, L. Cisneros and C. Dombrowski, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad, Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.
[41]	L. C. Wang, Y. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[42]	L. Wang and C. L. Mu, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.
[43]	Y. Wang and X. Cao, Global classical solutions of a 3d chemotaxis-Stokes system with rotation, Discrete Continuous Dynam. Systems - B, 20 (2015), 3235-3254. doi: 10.3934/dcdsb.2015.20.3235.
[44]	Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609. doi: 10.1016/j.jde.2015.08.027.
[45]	Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010.
[46]	Y. Wang and L. Xie, Boundedness for a 3D chemotaxis-Stokes system with porous medium diffusion and tensor-valued chemotaxis sensitivity. Z. Angew. Math. Phys., 68 (2017), p29. doi: 10.1007/s00033-017-0773-0.
[47]	Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780. doi: 10.1142/S0218202517500579.
[48]	M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.
[49]	M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319.
[50]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[51]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[52]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[53]	M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.
[54]	M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838.
[55]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[56]	M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modelling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.
[57]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.
[58]	M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9.
[59]	M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[60]	D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310. doi: 10.1016/j.na.2004.08.015.
[61]	C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signalling, J. Math. Biol., 70 (2015), 1-44. doi: 10.1007/s00285-013-0748-5.
[62]	C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math, 70 (2009), 133-167. doi: 10.1137/070711505.
[63]	Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012.
[64]	J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[65]	J. Zheng, Boundedness in a three-dimensional chemotaxis-fluid system involving tensor-valued sensitivity with saturation, J. Math. Anal. Appl., 442 (2016), 353-375. doi: 10.1016/j.jmaa.2016.04.047.
[66]	J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888. doi: 10.1016/j.jmaa.2015.05.071.
[67]	J. Zheng, A new approach toward locally bounded global solutions to a 3D chemotaxis-Stokes system with nonlinear diffusion and rotation, preprint, arXiv: 1701.01334.
[68]	J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629. doi: 10.1016/j.jde.2017.04.005.

show all references

References:

[1]	K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027.
[2]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[3]	X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913. doi: 10.1088/0951-7715/27/8/1899.
[4]	T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.
[5]	T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045.
[6]	T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004.
[7]	T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.
[8]	K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[9]	K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045.
[10]	M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
[11]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[12]	T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 281-301. doi: 10.1006/aama.2001.0721.
[13]	D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[14]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.- Ver., 105 (2003), 103-165.
[16]	S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2d bounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463.
[17]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
[18]	S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569.
[19]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[20]	A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858.
[21]	A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879.
[22]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[23]	H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106.
[24]	T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746. doi: 10.1142/S0218202515500177.
[25]	X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910. doi: 10.4310/CMS.2016.v14.n7.a5.
[26]	X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30. doi: 10.1016/j.nonrwa.2017.02.005.
[27]	J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028.
[28]	J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999. doi: 10.1016/j.jde.2016.03.030.
[29]	T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[30]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[31]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[32]	Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Zeitschrift f/"ur angewandte Mathematik und Physik, 68 (2017), p68. doi: 10.1007/s00033-017-0816-6.
[33]	Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
[34]	Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift f/"ur angewandte Mathematik und Physik, 67 (2016), p138. doi: 10.1007/s00033-016-0732-1.
[35]	Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y.
[36]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[37]	Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.
[38]	Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.
[39]	M. Winkler, Global weak solution in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.
[40]	I. Tuval, L. Cisneros and C. Dombrowski, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad, Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.
[41]	L. C. Wang, Y. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[42]	L. Wang and C. L. Mu, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.
[43]	Y. Wang and X. Cao, Global classical solutions of a 3d chemotaxis-Stokes system with rotation, Discrete Continuous Dynam. Systems - B, 20 (2015), 3235-3254. doi: 10.3934/dcdsb.2015.20.3235.
[44]	Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609. doi: 10.1016/j.jde.2015.08.027.
[45]	Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010.
[46]	Y. Wang and L. Xie, Boundedness for a 3D chemotaxis-Stokes system with porous medium diffusion and tensor-valued chemotaxis sensitivity. Z. Angew. Math. Phys., 68 (2017), p29. doi: 10.1007/s00033-017-0773-0.
[47]	Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780. doi: 10.1142/S0218202517500579.
[48]	M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.
[49]	M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319.
[50]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[51]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[52]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[53]	M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.
[54]	M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838.
[55]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[56]	M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modelling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.
[57]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.
[58]	M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487. doi: 10.1007/s00205-013-0678-9.
[59]	M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[60]	D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310. doi: 10.1016/j.na.2004.08.015.
[61]	C. Xue, Macroscopic equations for bacterial chemotaxis: Integration of detailed biochemistry of cell signalling, J. Math. Biol., 70 (2015), 1-44. doi: 10.1007/s00285-013-0748-5.
[62]	C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math, 70 (2009), 133-167. doi: 10.1137/070711505.
[63]	Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012.
[64]	J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[65]	J. Zheng, Boundedness in a three-dimensional chemotaxis-fluid system involving tensor-valued sensitivity with saturation, J. Math. Anal. Appl., 442 (2016), 353-375. doi: 10.1016/j.jmaa.2016.04.047.
[66]	J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888. doi: 10.1016/j.jmaa.2015.05.071.
[67]	J. Zheng, A new approach toward locally bounded global solutions to a 3D chemotaxis-Stokes system with nonlinear diffusion and rotation, preprint, arXiv: 1701.01334.
[68]	J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629. doi: 10.1016/j.jde.2017.04.005.

[1]	Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[2]	Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216
[3]	Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
[4]	Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[5]	Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[6]	Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[7]	Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[8]	Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
[9]	Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
[10]	Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[11]	Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
[12]	Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155
[13]	Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
[14]	Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[15]	Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035
[16]	Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026
[17]	Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
[18]	Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
[19]	Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[20]	Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069

American Institute of Mathematical Sciences